Existence and Asymptotic Behavior of Solutions for Generalized Choquard Systems

This paper concerns the linearly coupled system of nonlinear generalized Choquard equations in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N$$\end{document}: -Δφ+φ=(Aθ(x)⋆G(φ))G′(φ)+κψinRN,-Δψ+ψ=(Aθ(x)⋆H(ψ))H′(ψ)+κφinRN,φ(x),ψ(x)→0as|x|→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta \varphi +\varphi =\big (A_{\theta }(x)\star G(\varphi )\big )G'(\varphi )+\kappa \psi &{}\quad \text {in}\; {\mathbb {R}}^N, \\ -\Delta \psi + \psi =\big (A_{\theta }(x)\star H(\psi )\big )H'(\psi )+\kappa \varphi &{}\quad \text {in}\; {\mathbb {R}}^N, \\ \varphi (x),\psi (x)\rightarrow 0&{}\quad \text {as}\; |x|\rightarrow \infty , \end{array}\right. \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}, κ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa >0$$\end{document} is a coupling parameter, Aθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\theta }$$\end{document} is a Riesz potential, G,H∈C1(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G,H\in C^{1}({\mathbb {R}},{\mathbb {R}})$$\end{document}. Under almost necessary assumptions on G and H, we obtain a positive solution for the above system by using variational arguments. We also study the asymptotic behavior of the solution when κ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa \rightarrow 0$$\end{document}.